Functions applied from the right

Question

In some of the older books by Nathan Jacobson (like Lie Algebras and Lectures in Abstract Algebra), a convention is used that is quite uncommon at least today: Functions are applied from the right. For instance, a map $\theta$ applied to an object $x$ is written $x\theta$, sometimes even $x^\theta$. I do indeed see the benefits of this notation: Functions are applied from left to right, which is a nice analogue to the notation $$ \theta\gamma\colon x\stackrel\theta\longmapsto x\theta\stackrel\gamma\longmapsto x\theta\gamma. $$ However, it does not make the notation any less difficult to get used to. I was wondering if this notation was ever very common, or if it is just some Jacobson thing? Has anyone seen it elsewhere, and do you know its origin (if it has one)?

Answer 1

The use of functions on the right i.e. f(x) = xf is fairly common among algebracists. It confused the crap out of me in honors algebra when I studied out of I.R. Herstien's Topics in Algebra. (It really hurt me in that course because the teacher wrote functions on the left in his notes and never explained the difference.By the time I caught on to the fact the functions in the text were written on the right, my grade was damaged.So I have a bit a personal annoyance with it to be honest.) The thinking among algebracists is that it makes composition order of functions simpler to understand and since they deal with compositions on a regular basis, this is rather important. Consider g(f(h(x))). Writing it on the right: (((x)h)f)g

does seem to make the order easier to work with. But for those of us who learned the "calculus" notation of writing functions on the left, it's confusing as hell, especially at first.

Answer 2

In coding theory, codewords are often expressed as row vectors. Mappings are sometimes done by matrices. So the input (row vector) is on the left and the function (matrix) is on the right. $\vec{v}M$.

It takes a bit of getting used to from a standard linear algebra course where usually one works with column vectors, and linear transformations (matrices) on the left.